Motivation

For magnetic materials, a reasonable approximation of the the exact Helmholtz free energy, which fully characterizes all thermodynamic properties, is the separation into different contributions dominating on different time scales:

For a detailed description on the incorporation of the vibronic and electronic contributions we refer to our recent research project.

In this project we focus on the magnetic contribution F^mag for the free energy.

The most prominent magnetic material and basic ingredient of steels, iron, shows under ambient pressure two structural phase transitions with increasing temperature (bcc&rarr fcc&rarr bcc).
Although it is well established that magnetic excitations are of crucial importance for the structural phase stability and the structural transitions respectively [1,2], parameter free theories quantitatively predicting the thermondynamics of iron are still rare. Especially the quantitative impact of the different contributions (vibronic, electronic, magnetic) on the phase transitions has not been explained until now.
For modern steel research and ab initio based material design, the development of methods incorporating the magnetic free energy is therefore indispensable. In the past, several methods (e.g., a multi-band Hubbard model [3], spin fluctuation theory [4], the Heisenberg model [5]) have been used for calculating quantitative temperature dependent magnetic properties from ab initio input. However, most of them focused on the calculation of the magnetization and the Curie temperature and much less attention is paid to the magnetic free energy.
An existing approach based on the single-band spin-fluctuation theory within the mean field approximation gives a good qualitative agreement with experiment, but fails to provide a reasonable quantitative description of the magnetic free energy [2].

We have therefore developed a new parameter free methodology [6] based on the Heisenberg model to incorporate the magnetic free energy into the full Helmholtz energy (1). The results clearly show the necessity of taking all excitations into account (vibronic, electronic and magnetic) for correctly describing the experimental observations.

Theory

For systems having stable local magnetic moments as bcc iron [7], the Heisenberg model

has been successfully applied for describing thermodynamic properties [5,6,7,8,9].

The free energy and the heat capacity can directly be calculated via the internal energy U = ⟨H⟩:

: Fig. 1: Ab initio magnon spectrum of ferromagnetic bcc iron. The results are compared with available neutron scattering data. [13]

An elegant and efficient method to extract the exchange parameters J_ij from ab initio is the so called 'frozen magnon approach' [5]. Total energy calculations for different spin spirals are mapped onto the Hamiltonian (2). The difference of these total energy calculations can directly be attributed to the magnon frequencies in the system [10] (see Fig. 1 on the right).

With respect to the underlying model, the Tjablikov or often called Random Phase approximation [11] turned out to be a reasonable approximation. However, for general systems (S>1/2) the RPA shows inconsistencies regarding the decoupling for the transversal ⟨S₊S_-⟩ and longitudinal correlation function ⟨S_zS_z'⟩ [12], whereas for the special case S=1/2 a theory has already been worked out [11].

To simulate bcc iron (m=2.2 ⇒ S ≈ 1.1) we therefore bridge the different spin quantum systems by introducing a scaling factor based on the S=1/2-theory [6]:

Using the above formulation for the internal energy, the free energy (1) and the heat capacity (4) can be calculated.

Results

Using our new developed integrated approach combining all relevant excitation processes within the Helmholtz free energy, for the first time a quantitative description of the full free energy surface was possible [6]. More important, the quantitative impact of the different contributions can now be separated, see Fig. 2.

A sensitive quantity for the accuracy of the prediction of the free energy F(T) (1), is as its second derivative, the heat capacity C(T) [Eq. (4)]. Using the conventional meanfield approximation of the Hamiltonian (2), the strong increase of C(T) at the phase transition cannot be reproduced and the Curie temperature is overestimated by nearly 50 %. Within the RPA, the results are in good agreement with the experiment and reproduce also quantitatively the observed behavior at the critical temperature.

: Fig. 2: a) The free energy (1) versus experimental data (CalPhad). The different contributions to the free energy are highlighted using different colored areas (vibronic=yellow, electronic=green, magnetic=red). b) Heat capacity a function of temperature including the various contributions (1). The magnetic contribution is indicated by the red colored area. A simple Meanfield solution of (2) is not sufficient (red dashed line).

References

[1] L. Kaufman and H. Bernstein, Computer Calculation of Phase Diagrams (Academic, New York, 1970).
[2] H. Hasegawa and D. G. Pettifor, Phys. Rev. Lett. 50, 130 (1983).
[3] W. Nolting, A. Vega, and T. Fauster, Z. Phys. B 96, 357 (1995).
[4] J. Kübler, G. H. Fecher, and C. Felser, Phys. Rev. B 76, 024414 (2007).
[5] S. V. Halilov, A. Y. Perlov, P. M. Oppeneer, and H. Eschrig, Europhys. Lett. 39, 91 (1997).
[6] F. Körmann, A. Dick, B. Grabowski, B. Hallstedt, T. Hickel, and J. Neugebauer, Phys. Rev. B 78, 033102 (2008).
[7] A. V. Ruban, S. Khmelevskyi, P. Mohn, and B. Johansson, Phys. Rev. B 75, 054402 (2007).
[8] F. Körmann, A. Dick, T. Hickel, and J. Neugebauer, Phys. Rev. B 79, 184406 (2009).
[9] T. Hickel, A. Dick, B. Grabowski, F. Körmann, J. Neugebauer, Steel Res. Int. 80, 4 (2009).[10] J. Kübler, J. Phys.: Condens. Matter 18, 9795 (2006).
[11] S. V. Tjablikov, in Methods in the quantum theory of magnetism (Plenum Press, New York, 1967).
[12] R. A. Tahir-Kheli, Phys. Rev. 159, 439 (1967).
[13] J. Lynn, Phys. Rev. B 11, 2624 (1975); C. Loong, J. Carpenter, J. Lynn, R. Robinson, and H. Mook, J. Appl. Phys. 55, 1895 (1984).

Ab initio up to the melting point

Multi-physics modeling of strain-induced interactions in interstitial solid solutions

This page is maintained by Blazej Grabowski. Last update: 25.03.2010